Diophantine Equations and Inequalities in Algebraic Number Fields

by Yuan Wang

Overview

1. The Circle Method and Waring's Problem.- 1.1 Introduction.- 1.2 Farey Division.- 1.3 Auxiliary Lemmas.- 1.4 Major Arcs.- 1.5 Singular Integral.- 1.6 Singular Series.- 1.7 Proof of Lemma 1.12.- 1.8 Proof of Theorem 1.1.- Notes.- 2. Complete Exponential Sums.- 2.1 Introduction.- 2.2 Several Lemmas.- 2.3 Mordell's Lemma.- 2.4 Fundamental Lemma.- 2.5 Proof of Theorem 2.1.- 2.6 Proof of Theorem 2.2.- Notes.- 3. Weyl's Sums.- 3.1 Introduction.- 3.2 Proof of Theorem 3.1.- 3.3 A Lemma on Units.- 3.4 The Asymptotic Formula for N(a, T).- 3.5 A Sum.- 3.6 Mitsui's Lemma.- 3.7 Proof of Theorem 3.3.- 3.8 Proof of Lemma 3.6.- 3.9 Continuation.- Notes.- 4. Mean Value Theorems.- 4.1 Introduction.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 4.4 A Lemma on the Set D.- 4.5 A Lemma on the Set D(x).- 4.6 Fundamental Lemma.- 4.7 Proof of Lemma 4.1.- Notes.- 5. The Circle Method in Algebraic Number Fields.- 5.1 Introduction.- 5.2 Lemmas.- 5.3 Asympotic Expansion forSi (?, T).- 5.4 Further Estimates on Basic Domains.- 5.5 Proof of Theorem 5.1.- 5.6 Proof of Theorem 5.2.- Notes.- 6. Singular Series and Singular Integrals.- 6.1 Introduction.- 6.2 Product Form for Singular Series.- 6.3 Singular Series and Congruences.- 6.4 p-adic Valuations.- 6.5 k-th Power Residues.- 6.6 Proof of Theorem 6.1.- 6.7 Monotonic Functions.- 6.8 Proof of Theorem 6.2.- Notes.- 7. Waring's Problem.- 7.1 Introduction.- 7.2 The Ring Jk.- 7.3 Proofs of Theorems 7.1 and 7.2.- 7.4 Proof of Theorem 7.3.- 7.5 Proof of Theorem 7.4.- Notes.- 8. Additive Equations.- 8.1 Introduction.- 8.2 Reductions.- 8.3 Contraction.- 8.4 Derived Variables.- 8.5 Proof of Theorem 8.1.- 8.6 Proof of Theorem 8.2.- 8.7 Bounds for Solutions.- Notes.- 9. Small Nonnegative Solutions of Additive Equations.- 9.1 Introduction.- 9.2 Hurwitz's Lemma.- 9.3 Reductions.- 9.4 Continuation.- 9.5 Farey Division.- 9.6 Supplementary Domain.- 9.7 Basic Domains.- 9.8 Proof of Theorem 9.1.- Notes.- 10. Small Solutions of Additive Equations.- 10.1 Introduction.- 10.2 Reductions.- 10.3 Continuation.- 10.4 Farey Division.- 10.5 Supplementary Domain.- 10.6 Basic Domains.- 10.7 Proof of Theorem 10.1.- Notes.- 11. Diophantine Inequalities for Forms.- 11.1 Introduction.- 11.2 A Single Additive Form.- 11.3 A Variant Circle Method.- 11.4 Continuation.- 11.5 Proof of Lemma 11.1.- 11.6 Linear Forms.- 11.7 A Single Form.- 11.8 Proof of Theorem 11.1.- Notes.- References I.- References II

Product Details

ISBN-13:	9783540520191
Publisher:	Springer-Verlag New York, LLC
Publication date:	03/28/1991
Pages:	192
Product dimensions:	6.69(w) x 9.61(h) x 0.50(d)

Table of Contents

1. The Circle Method and Waring’s Problem.- 1.1 Introduction.- 1.2 Farey Division.- 1.3 Auxiliary Lemmas.- 1.4 Major Arcs.- 1.5 Singular Integral.- 1.6 Singular Series.- 1.7 Proof of Lemma 1.12.- 1.8 Proof of Theorem 1.1.- Notes.- 2. Complete Exponential Sums.- 2.1 Introduction.- 2.2 Several Lemmas.- 2.3 Mordell’s Lemma.- 2.4 Fundamental Lemma.- 2.5 Proof of Theorem 2.1.- 2.6 Proof of Theorem 2.2.- Notes.- 3. Weyl’s Sums.- 3.1 Introduction.- 3.2 Proof of Theorem 3.1.- 3.3 A Lemma on Units.- 3.4 The Asymptotic Formula for N(a,T).- 3.5 A Sum.- 3.6 Mitsui’s Lemma.- 3.7 Proof of Theorem 3.3.- 3.8 Proof of Lemma 3.6.- 3.9 Continuation.- Notes.- 4. Mean Value Theorems.- 4.1 Introduction.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 4.4 A Lemma on the Set D.- 4.5 A Lemma on the Set D(x).- 4.6 Fundamental Lemma.- 4.7 Proof of Lemma 4.1.- Notes.- 5. The Circle Method in Algebraic Number Fields.- 5.1 Introduction.- 5.2 Lemmas.- 5.3 Asympotic Expansion forSi (?, T).- 5.4 Further Estimates on Basic Domains.- 5.5 Proof of Theorem 5.1.- 5.6 Proof of Theorem 5.2.- Notes.- 6. Singular Series and Singular Integrals.- 6.1 Introduction.- 6.2 Product Form for Singular Series.- 6.3 Singular Series and Congruences.- 6.4 p-adic Valuations.- 6.5 k-th Power Residues.- 6.6 Proof of Theorem 6.1.- 6.7 Monotonic Functions.- 6.8 Proof of Theorem 6.2.- Notes.- 7. Waring’s Problem.- 7.1 Introduction.- 7.2 The Ring Jk.- 7.3 Proofs of Theorems 7.1 and 7.2.- 7.4 Proof of Theorem 7.3.- 7.5 Proof of Theorem 7.4.- Notes.- 8. Additive Equations.- 8.1 Introduction.- 8.2 Reductions.- 8.3 Contraction.- 8.4 Derived Variables.- 8.5 Proof of Theorem 8.1.- 8.6 Proof of Theorem 8.2.- 8.7 Bounds for Solutions.- Notes.- 9. Small Nonnegative Solutions of Additive Equations.- 9.1 Introduction.- 9.2 Hurwitz’s Lemma.- 9.3 Reductions.- 9.4 Continuation.- 9.5 Farey Division.- 9.6 Supplementary Domain.- 9.7 Basic Domains.- 9.8 Proof of Theorem 9.1.- Notes.- 10. Small Solutions of Additive Equations.- 10.1 Introduction.- 10.2 Reductions.- 10.3 Continuation.- 10.4 Farey Division.- 10.5 Supplementary Domain.- 10.6 Basic Domains.- 10.7 Proof of Theorem 10.1.- Notes.- 11. Diophantine Inequalities for Forms.- 11.1 Introduction.- 11.2 A Single Additive Form.- 11.3 A Variant Circle Method.- 11.4 Continuation.- 11.5 Proof of Lemma 11.1.- 11.6 Linear Forms.- 11.7 A Single Form.- 11.8 Proof of Theorem 11.1.- Notes.- References I.- References II.

Editorial Reviews

Wang gives generalizations of important results on diophantine equations and inequalities over rational fields to algebraic number fields. He also offers an account of Siegal's generalized circle method and its applications to Waring's problem and additive equations and an account of Schmidt's method on diophantine equations and inequalities in several variables in algebraic number fields. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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